This book is concerned with the quantitative aspects of the theory of nonlinear diffusion equations; equations which can be seen as nonlinear variations of the classical heat equation. They appear as ...
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This book is concerned with the quantitative aspects of the theory of nonlinear diffusion equations; equations which can be seen as nonlinear variations of the classical heat equation. They appear as mathematical models in different branches of physics, chemistry, biology, and engineering, and are also relevant in differential geometry and relativistic physics. Much of the modern theory of such equations is based on estimates and functional analysis. Concentrating on a class of equations with nonlinearities of power type that lead to degenerate or singular parabolicity (equations of porous medium type), the aim of this book is to obtain sharp a priori estimates and decay rates for general classes of solutions in terms of estimates of particular problems. These estimates are the building blocks in understanding the qualitative theory, and the decay rates pave the way to the fine study of asymptotics. Many technically relevant questions are presented and analyzed in detail. A systematic picture of the most relevant phenomena is obtained for the equations under study, including time decay, smoothing, extinction in finite time, and delayed regularity.Less

Juan Luis Vázquez

Published in print: 2006-08-03

This book is concerned with the quantitative aspects of the theory of nonlinear diffusion equations; equations which can be seen as nonlinear variations of the classical heat equation. They appear as mathematical models in different branches of physics, chemistry, biology, and engineering, and are also relevant in differential geometry and relativistic physics. Much of the modern theory of such equations is based on estimates and functional analysis. Concentrating on a class of equations with nonlinearities of power type that lead to degenerate or singular parabolicity (equations of porous medium type), the aim of this book is to obtain sharp a priori estimates and decay rates for general classes of solutions in terms of estimates of particular problems. These estimates are the building blocks in understanding the qualitative theory, and the decay rates pave the way to the fine study of asymptotics. Many technically relevant questions are presented and analyzed in detail. A systematic picture of the most relevant phenomena is obtained for the equations under study, including time decay, smoothing, extinction in finite time, and delayed regularity.

This chapter discusses Yaari's (19665) annuitization result, which states that a risk-averse individual concerned about longevity risk (uncertain length of life) will always purchase actuarially-fair ...
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This chapter discusses Yaari's (19665) annuitization result, which states that a risk-averse individual concerned about longevity risk (uncertain length of life) will always purchase actuarially-fair annuity contracts, enabling them to smooth consumption in every period of retirement. The chapter explains the assumptions behind this result. It models the demand for annuities in an expected utility framework, and demonstrates the value of annuities under various specifications of preferences.Less

Annuity demand theory

Edmund CannonIan Tonks

Published in print: 2008-10-02

This chapter discusses Yaari's (19665) annuitization result, which states that a risk-averse individual concerned about longevity risk (uncertain length of life) will always purchase actuarially-fair annuity contracts, enabling them to smooth consumption in every period of retirement. The chapter explains the assumptions behind this result. It models the demand for annuities in an expected utility framework, and demonstrates the value of annuities under various specifications of preferences.

Several recent advances in smoothing and semiparametric regression are presented in this book from a unifying, Bayesian perspective. Simulation-based full Bayesian Markov chain Monte Carlo (MCMC) ...
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Several recent advances in smoothing and semiparametric regression are presented in this book from a unifying, Bayesian perspective. Simulation-based full Bayesian Markov chain Monte Carlo (MCMC) inference, as well as empirical Bayes procedures closely related to penalized likelihood estimation and mixed models, are considered here. Throughout, the focus is on semiparametric regression and smoothing based on basis expansions of unknown functions and effects in combination with smoothness priors for the basis coefficients. Beginning with a review of basic methods for smoothing and mixed models, longitudinal data, spatial data, and event history data are treated in separate chapters. Worked examples from various fields such as forestry, development economics, medicine, and marketing are used to illustrate the statistical methods covered in this book. Most of these examples have been analysed using implementations in the Bayesian software, BayesX, and some with R Codes.Less

Bayesian Smoothing and Regression for Longitudinal, Spatial and Event History Data

Ludwig FahrmeirThomas Kneib

Published in print: 2011-04-28

Several recent advances in smoothing and semiparametric regression are presented in this book from a unifying, Bayesian perspective. Simulation-based full Bayesian Markov chain Monte Carlo (MCMC) inference, as well as empirical Bayes procedures closely related to penalized likelihood estimation and mixed models, are considered here. Throughout, the focus is on semiparametric regression and smoothing based on basis expansions of unknown functions and effects in combination with smoothness priors for the basis coefficients. Beginning with a review of basic methods for smoothing and mixed models, longitudinal data, spatial data, and event history data are treated in separate chapters. Worked examples from various fields such as forestry, development economics, medicine, and marketing are used to illustrate the statistical methods covered in this book. Most of these examples have been analysed using implementations in the Bayesian software, BayesX, and some with R Codes.

This chapter describes the modulus of smoothness of a function in the direction of a family of subspaces and the much simpler notion of upper Fréchet differentiability. It also considers the notion ...
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This chapter describes the modulus of smoothness of a function in the direction of a family of subspaces and the much simpler notion of upper Fréchet differentiability. It also considers the notion of spaces admitting bump functions smooth in the direction of a family of subspaces with modulus controlled by ω‎(t). It shows that this notion is related to asymptotic uniform smoothness, and that very smooth bumps, and very asymptotically uniformly smooth norms, exist in all asymptotically c₀ spaces. This allows a new approach to results on Γ‎-almost everywhere Frechet differentiability of Lipschitz functions. The chapter concludes by explaining an immediate consequence for renorming of spaces containing an asymptotically c₀ family of subspaces.Less

Smoothness and Asymptotic Smoothness

Joram LindenstraussDavid PreissTiˇser Jaroslav

Published in print: 2012-02-26

This chapter describes the modulus of smoothness of a function in the direction of a family of subspaces and the much simpler notion of upper Fréchet differentiability. It also considers the notion of spaces admitting bump functions smooth in the direction of a family of subspaces with modulus controlled by ω‎(t). It shows that this notion is related to asymptotic uniform smoothness, and that very smooth bumps, and very asymptotically uniformly smooth norms, exist in all asymptotically c₀ spaces. This allows a new approach to results on Γ‎-almost everywhere Frechet differentiability of Lipschitz functions. The chapter concludes by explaining an immediate consequence for renorming of spaces containing an asymptotically c₀ family of subspaces.

Business and Management, Public Management, Pensions and Pension Management

Unless defined benefit pension plans are managed much better and more cost-effectively, they will be replaced by defined contribution plans. Benefit and contribution policies need to be carefully ...
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Unless defined benefit pension plans are managed much better and more cost-effectively, they will be replaced by defined contribution plans. Benefit and contribution policies need to be carefully evaluated to make sure that a reasonable level of ongoing contributions, together with investment income, are adequate to fund the defined benefit plan without unpleasant surprises. Unless valuation and contribution conventions change to market-valued economically based quantities, decision makers will lack the right information with which to make informed policy decisions.Less

Between Scylla and Charybdis: Improving the Cost Effectiveness of Public Pension Retirement Plans

M. Barton Waring

Published in print: 2009-08-13

Unless defined benefit pension plans are managed much better and more cost-effectively, they will be replaced by defined contribution plans. Benefit and contribution policies need to be carefully evaluated to make sure that a reasonable level of ongoing contributions, together with investment income, are adequate to fund the defined benefit plan without unpleasant surprises. Unless valuation and contribution conventions change to market-valued economically based quantities, decision makers will lack the right information with which to make informed policy decisions.

This chapter discusses spatial variation in risk. Epidemiological disease investigations should include an assessment of the spatial variation of disease risk, as this may provide important clues ...
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This chapter discusses spatial variation in risk. Epidemiological disease investigations should include an assessment of the spatial variation of disease risk, as this may provide important clues leading to causal explanations. The objective is to produce a map representation of the important spatial effects present in the data while simultaneously removing any distracting noise or extreme values. The resulting smoothed map should have increased precision without introducing significant bias. The method used to analyse the data depends on how they have been recorded. Smoothing based on kernel functions, smoothing based and on Bayesian models, and spatial interpolation are discussed.Less

Spatial variation in risk

Published in print: 2008-05-29

This chapter discusses spatial variation in risk. Epidemiological disease investigations should include an assessment of the spatial variation of disease risk, as this may provide important clues leading to causal explanations. The objective is to produce a map representation of the important spatial effects present in the data while simultaneously removing any distracting noise or extreme values. The resulting smoothed map should have increased precision without introducing significant bias. The method used to analyse the data depends on how they have been recorded. Smoothing based on kernel functions, smoothing based and on Bayesian models, and spatial interpolation are discussed.

Inferring the probability density function (pdf) from a sample of data is known as density estimation. The same methodology is often called data smoothing. Density estimation in the one-dimensional ...
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Inferring the probability density function (pdf) from a sample of data is known as density estimation. The same methodology is often called data smoothing. Density estimation in the one-dimensional case has been discussed in the previous chapters. This chapter extends it to multidimensional cases. Density estimation is one of the most critical components of extracting knowledge from data. For example, given a pdf estimated from point data, we can generate simulated distributions of data and compare them against observations. If we can identify regions of low probability within the pdf, we have a mechanism for the detection of unusual or anomalous sources. If our point data can be separated into subsamples using provided class labels, we can estimate the pdf for each subsample and use the resulting set of pdfs to classify new points: the probability that a new point belongs to each subsample/class is proportional to the pdf of each class evaluated at the position of the point.Less

Searching for Structure in Point Data

Published in print: 2014-01-12

Inferring the probability density function (pdf) from a sample of data is known as density estimation. The same methodology is often called data smoothing. Density estimation in the one-dimensional case has been discussed in the previous chapters. This chapter extends it to multidimensional cases. Density estimation is one of the most critical components of extracting knowledge from data. For example, given a pdf estimated from point data, we can generate simulated distributions of data and compare them against observations. If we can identify regions of low probability within the pdf, we have a mechanism for the detection of unusual or anomalous sources. If our point data can be separated into subsamples using provided class labels, we can estimate the pdf for each subsample and use the resulting set of pdfs to classify new points: the probability that a new point belongs to each subsample/class is proportional to the pdf of each class evaluated at the position of the point.

This chapter explores a set of biblical texts dealing with the loss or absence of hair: the tale of David's envoys in 2 Samuel 10:4–5; related texts concerning conquest and images of shaving; ...
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This chapter explores a set of biblical texts dealing with the loss or absence of hair: the tale of David's envoys in 2 Samuel 10:4–5; related texts concerning conquest and images of shaving; passages discussing mourning practices and other ritual passages; and finally the significant contrasts drawn by biblical writers between hairy and smooth men. The stories of Esau and Jacob, Elijah and Elisha, and Joseph as prisoner versus Joseph as servant of Pharaoh are discussed. Saul Olyan has explored a number these texts with insight, cautioning the reader to pay special attention to shaving in context. He notes, however, that many passages concerning the elimination of hair involve some sort of alteration in status, such as a return to a state of purity after a period of uncleanness or a marking of the death of a loved one and the reintegration to the realm of the living after the loved one's demise.Less

Absent Hair

Susan Niditch

Published in print: 2008-03-01

This chapter explores a set of biblical texts dealing with the loss or absence of hair: the tale of David's envoys in 2 Samuel 10:4–5; related texts concerning conquest and images of shaving; passages discussing mourning practices and other ritual passages; and finally the significant contrasts drawn by biblical writers between hairy and smooth men. The stories of Esau and Jacob, Elijah and Elisha, and Joseph as prisoner versus Joseph as servant of Pharaoh are discussed. Saul Olyan has explored a number these texts with insight, cautioning the reader to pay special attention to shaving in context. He notes, however, that many passages concerning the elimination of hair involve some sort of alteration in status, such as a return to a state of purity after a period of uncleanness or a marking of the death of a loved one and the reintegration to the realm of the living after the loved one's demise.

This book provides an understanding of how events at the cellular level impact on the cardiovascular system as a whole. Advances in knowledge are highlighted and all the themes are presented from the ...
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This book provides an understanding of how events at the cellular level impact on the cardiovascular system as a whole. Advances in knowledge are highlighted and all the themes are presented from the single cell (smooth muscle endothelial and nerve) level through to the blood vessel wall to the vascular system as a functional system. This book provides an introduction to wide-ranging pharmacological principles and major techniques in this subject area, and is a source of background literature for research in vascular pharmacology.Less

The Pharmacology of Vascular Smooth Muscle

Published in print: 1996-02-22

This book provides an understanding of how events at the cellular level impact on the cardiovascular system as a whole. Advances in knowledge are highlighted and all the themes are presented from the single cell (smooth muscle endothelial and nerve) level through to the blood vessel wall to the vascular system as a functional system. This book provides an introduction to wide-ranging pharmacological principles and major techniques in this subject area, and is a source of background literature for research in vascular pharmacology.

The concept of a differential equation controlled by a rough path can be motivated by quite simple examples. One such example is a linear system driven by two-dimensional noise. This example is ...
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The concept of a differential equation controlled by a rough path can be motivated by quite simple examples. One such example is a linear system driven by two-dimensional noise. This example is developed and an explicit answer is given to the question. Exactly what information should I extract from the driving stimulus or noise in order to accurately predict the response? The notions of a controlled system of Chen's iterated integral are introduced. The main notion in this book is the concept of a rough path. Almost all paths that one encounters in everyday life are only described approximately. Newton observed that a smooth path is actually quite well approximated by its chords. If one wants to describe a path γ over a short time interval from s to t, then it is enough to evaluate γ at these two times and consider the approximation that comes from replacing γ by the straight line with the same increment. This approach is not adequate if the control or path γ is oscillatory on the scale witnessed by the times s and t. If the path γ represented a text, then the chord is simply a word count. It turns out that a better description, one which takes into account the order of the events represented by γ, can be achieved by a description of γ that involves its first few Chen iterated integrals.Less

INTRODUCTION

Terry LyonsZhongmin Qian

Published in print: 2002-12-19

The concept of a differential equation controlled by a rough path can be motivated by quite simple examples. One such example is a linear system driven by two-dimensional noise. This example is developed and an explicit answer is given to the question. Exactly what information should I extract from the driving stimulus or noise in order to accurately predict the response? The notions of a controlled system of Chen's iterated integral are introduced. The main notion in this book is the concept of a rough path. Almost all paths that one encounters in everyday life are only described approximately. Newton observed that a smooth path is actually quite well approximated by its chords. If one wants to describe a path γ over a short time interval from s to t, then it is enough to evaluate γ at these two times and consider the approximation that comes from replacing γ by the straight line with the same increment. This approach is not adequate if the control or path γ is oscillatory on the scale witnessed by the times s and t. If the path γ represented a text, then the chord is simply a word count. It turns out that a better description, one which takes into account the order of the events represented by γ, can be achieved by a description of γ that involves its first few Chen iterated integrals.

This chapter considers the diffusion equation: an implicit in time discretization leads to the Helmholtz equation. Both the temporal discretization and eigenspectra of second-order operators that ...
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This chapter considers the diffusion equation: an implicit in time discretization leads to the Helmholtz equation. Both the temporal discretization and eigenspectra of second-order operators that dictate time-step restrictions are discussed. Appropriate preconditioning techniques for inversion of the stiffness matrix, non-smooth solutions due to geometric singularities, and recent advances in three-dimensional domains are discussed. The exercises at the end of the chapter build on the exercises of Chapters 3 and 4 to implement a two-dimensional standard Galerkin hp solution to the Helmholtz problem.Less

DIFFUSION EQUATION

George Em KarniadakisSpencer J. Sherwin

Published in print: 2005-06-02

This chapter considers the diffusion equation: an implicit in time discretization leads to the Helmholtz equation. Both the temporal discretization and eigenspectra of second-order operators that dictate time-step restrictions are discussed. Appropriate preconditioning techniques for inversion of the stiffness matrix, non-smooth solutions due to geometric singularities, and recent advances in three-dimensional domains are discussed. The exercises at the end of the chapter build on the exercises of Chapters 3 and 4 to implement a two-dimensional standard Galerkin hp solution to the Helmholtz problem.

This chapter studies the broad class of models which have difficult problems of nonlinearity and multiple roots. These models include the mixture models, a model for bivariate normal paired data with ...
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This chapter studies the broad class of models which have difficult problems of nonlinearity and multiple roots. These models include the mixture models, a model for bivariate normal paired data with standardized marginals, the stratified normal with common mean, regression with measurement error, weighted likelihood method, the Cauchy distribution, the symmetric stable laws, and examples with inconsistent global maximum likelihood estimates. The Tobit model is used to illustrate how one may determine if an estimating equation will have multiple solutions. The difficult issue of finding all possible roots is also discussed. This chapter also shows that multiple roots may be treated as a point process whose intensity may be assessed in the parameter space. Finally, smoothing is introduced as a technique to modify the estimating function or the artificial likelihood so as to remove unstable extraneous roots.Less

Working with roots

Christopher G. SmallJinfang Wang

Published in print: 2003-10-02

This chapter studies the broad class of models which have difficult problems of nonlinearity and multiple roots. These models include the mixture models, a model for bivariate normal paired data with standardized marginals, the stratified normal with common mean, regression with measurement error, weighted likelihood method, the Cauchy distribution, the symmetric stable laws, and examples with inconsistent global maximum likelihood estimates. The Tobit model is used to illustrate how one may determine if an estimating equation will have multiple solutions. The difficult issue of finding all possible roots is also discussed. This chapter also shows that multiple roots may be treated as a point process whose intensity may be assessed in the parameter space. Finally, smoothing is introduced as a technique to modify the estimating function or the artificial likelihood so as to remove unstable extraneous roots.

In this chapter we review recent advances in numerical simulation of micro and nanoflows. For coarse-grained simulation of microfuidics, we present an overview of Lattice Boltzmann, Brownian ...
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In this chapter we review recent advances in numerical simulation of micro and nanoflows. For coarse-grained simulation of microfuidics, we present an overview of Lattice Boltzmann, Brownian dynamics, stochastic rotation dynamics, and smoothed particle hydrodynamics methods and discuss the dissipative particle dynamics method in detail as it shares many features with the other methods. In the area of nanoflows, we review recent advances in non-equilibrium molecular dynamics methods focusing on the development of self-consistent and grand canonical methods for electric-field mediated transport. We present examples showing the significance of quantum effects in nanoflows. Finally, we discuss multiscale modeling focusing on direct coupling of molecular dynamics with Navier-Stokes equations and hierarchical coupling of quantum, molecular dynamics and classical fluid equations.Less

Numerical Simulation Of Microflows And Nanoflows

Narayan R. AluruGeorge Em Karniadakis

Published in print: 2010-03-25

In this chapter we review recent advances in numerical simulation of micro and nanoflows. For coarse-grained simulation of microfuidics, we present an overview of Lattice Boltzmann, Brownian dynamics, stochastic rotation dynamics, and smoothed particle hydrodynamics methods and discuss the dissipative particle dynamics method in detail as it shares many features with the other methods. In the area of nanoflows, we review recent advances in non-equilibrium molecular dynamics methods focusing on the development of self-consistent and grand canonical methods for electric-field mediated transport. We present examples showing the significance of quantum effects in nanoflows. Finally, we discuss multiscale modeling focusing on direct coupling of molecular dynamics with Navier-Stokes equations and hierarchical coupling of quantum, molecular dynamics and classical fluid equations.

This chapter addresses the question of extending the theory and estimates of the FDE to cover the singular range m ≤ 0, that has only been partially considered in previous chapters. This completes ...
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This chapter addresses the question of extending the theory and estimates of the FDE to cover the singular range m ≤ 0, that has only been partially considered in previous chapters. This completes the scope of the investigation about decay, smoothing effects, and best constants. But the range offers the possibility of learning about a novel and quite interesting dynamical issue: instantaneous extinction, which is tied to boundary layers at the initial time. The second part of the chapter discusses the question of local estimates improving on the results of previous sections. Finally an open problem is presented.Less

Superfast FDE

Juan Luis Vázquez

Published in print: 2006-08-03

This chapter addresses the question of extending the theory and estimates of the FDE to cover the singular range m ≤ 0, that has only been partially considered in previous chapters. This completes the scope of the investigation about decay, smoothing effects, and best constants. But the range offers the possibility of learning about a novel and quite interesting dynamical issue: instantaneous extinction, which is tied to boundary layers at the initial time. The second part of the chapter discusses the question of local estimates improving on the results of previous sections. Finally an open problem is presented.

This chapter presents the application of the foregoing ideas to the p-Laplacian evolution equation and the doubly nonlinear evolution equation. Topics covered include the doubly linear diffusion ...
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This chapter presents the application of the foregoing ideas to the p-Laplacian evolution equation and the doubly nonlinear evolution equation. Topics covered include the doubly linear diffusion equation, symmetrization and mass comparison, source-type solutions, doubly nonlinear diffusion equation, and smoothing estimates, best constants, and decay rates for PLE and DNLE.Less

Evolution equations of the p-Laplacian type

Juan Luis Vázquez

Published in print: 2006-08-03

This chapter presents the application of the foregoing ideas to the p-Laplacian evolution equation and the doubly nonlinear evolution equation. Topics covered include the doubly linear diffusion equation, symmetrization and mass comparison, source-type solutions, doubly nonlinear diffusion equation, and smoothing estimates, best constants, and decay rates for PLE and DNLE.

This introductory chapter opens with an explanation of the purpose of the book, which is to obtain basic estimates for some particular nonlinear parabolic equations and to derive consequences about ...
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This introductory chapter opens with an explanation of the purpose of the book, which is to obtain basic estimates for some particular nonlinear parabolic equations and to derive consequences about qualitative and quantitative aspects of the theory. The technical tools used, goals, contents and distribution, and how to read and use the book are discussed.Less

Introduction

Juan Luis Vázquez

Published in print: 2006-08-03

This introductory chapter opens with an explanation of the purpose of the book, which is to obtain basic estimates for some particular nonlinear parabolic equations and to derive consequences about qualitative and quantitative aspects of the theory. The technical tools used, goals, contents and distribution, and how to read and use the book are discussed.

This chapter discusses the smoothing and decay effects for the porous medium equation, using as a model case the famous Barenblatt solutions that have explicit formulas. Topics covered include ...
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This chapter discusses the smoothing and decay effects for the porous medium equation, using as a model case the famous Barenblatt solutions that have explicit formulas. Topics covered include source-type solutions, smoothing effect and decay with L 1 functions or measures as data, smoothing exponents and scaling properties, strong and weak smoothing effects, comparison for different diffusivities, a general smoothing result, and estimating the smoothing effect into Lp .Less

Smoothing effect and time decay. Data in L 1 (R n ) or M (R n )

Juan Luis Vázquez

Published in print: 2006-08-03

This chapter discusses the smoothing and decay effects for the porous medium equation, using as a model case the famous Barenblatt solutions that have explicit formulas. Topics covered include source-type solutions, smoothing effect and decay with L1 functions or measures as data, smoothing exponents and scaling properties, strong and weak smoothing effects, comparison for different diffusivities, a general smoothing result, and estimating the smoothing effect into Lp.

This chapter addresses the question of boundedness for the same equation when the initial data are chosen in the Lebesgue space Lp, p Ɛ (1, ∞). It is assumed that m > mc . The ...
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This chapter addresses the question of boundedness for the same equation when the initial data are chosen in the Lebesgue space Lp, p Ɛ (1, ∞). It is assumed that m > mc . The results are based on a very delicate phase-plane analysis of the existence of certain types of self-similar solutions. This technique will play a big role in later chapters and the analysis is presented in Section 3.2. The technique allows us to extend the functional setting in a natural way from the Lebesgue spaces into the Marcinkiewicz spaces Mp (R n ). Using this machinery, a special solution is developed in Section 3.3 that replaces the ZKB in the present context and allows us to establish the smoothing effect from Mp into L ∞ in Section 3.4. This effect is easily extended into a similar effect that takes place.Less

Smoothing effect and time decay from L p or M p

Juan Luis Vázquez

Published in print: 2006-08-03

This chapter addresses the question of boundedness for the same equation when the initial data are chosen in the Lebesgue space Lp, p Ɛ (1, ∞). It is assumed that m > mc. The results are based on a very delicate phase-plane analysis of the existence of certain types of self-similar solutions. This technique will play a big role in later chapters and the analysis is presented in Section 3.2. The technique allows us to extend the functional setting in a natural way from the Lebesgue spaces into the Marcinkiewicz spaces Mp(Rn). Using this machinery, a special solution is developed in Section 3.3 that replaces the ZKB in the present context and allows us to establish the smoothing effect from Mp into L∞ in Section 3.4. This effect is easily extended into a similar effect that takes place.

This chapter studies the smoothing effect and decay rates for the FDE in the subcritical range m < mc , and also for the critical exponent mc = (n - 2)/n. ...
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This chapter studies the smoothing effect and decay rates for the FDE in the subcritical range m < mc , and also for the critical exponent mc = (n - 2)/n. While advancing some of the results, which are valid for the whole subcritical range m < mc , it focuses on the case m > 0. The chapter is organized as follows. Section 5.2 contains the proof of extinction of solutions in the Marcinkiewicz space Mp* (R n), that is characterized as the natural extinction space among all the spaces Mp (R n) and Lp (R n). Section 5.3 considers the question of necessary conditions and the continuity of the extinction time T as a function of u0. Section 5.4 deals with the construction of the global self-similar solutions that increase their space decay rate for positive time. Section 5.5 explains how and where whole mass is lost in the process of extinction. Section 5.6 studies forward effects when dealing with the critical exponent m = mc with starting space L1 . Finally, Section 5.7 discusses extinction as a form of blow-up after a suitable change of variables.Less

Juan Luis Vázquez

Published in print: 2006-08-03

This chapter studies the smoothing effect and decay rates for the FDE in the subcritical range m < mc, and also for the critical exponent mc = (n - 2)/n. While advancing some of the results, which are valid for the whole subcritical range m < mc, it focuses on the case m > 0. The chapter is organized as follows. Section 5.2 contains the proof of extinction of solutions in the Marcinkiewicz space Mp* (Rn), that is characterized as the natural extinction space among all the spaces Mp (Rn) and Lp (Rn). Section 5.3 considers the question of necessary conditions and the continuity of the extinction time T as a function of u0. Section 5.4 deals with the construction of the global self-similar solutions that increase their space decay rate for positive time. Section 5.5 explains how and where whole mass is lost in the process of extinction. Section 5.6 studies forward effects when dealing with the critical exponent m = mc with starting space L1. Finally, Section 5.7 discusses extinction as a form of blow-up after a suitable change of variables.

This chapter studies two transition situations where non-uniqueness plays an important role. The first deals with the range -1 < m ≤ 0 in n = 1, which looks like supercritical but contains the ...
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This chapter studies two transition situations where non-uniqueness plays an important role. The first deals with the range -1 < m ≤ 0 in n = 1, which looks like supercritical but contains the non-uniqueness phenomenon for the Cauchy problem. The second is the study of logarithmic diffusion in the plane, i.e., the case m = 0 for n = 2 which has many appealing features for the analyst and the geometer.Less

Logarithmic diffusion in 2D and intermediate 1D range

Juan Luis Vázquez

Published in print: 2006-08-03

This chapter studies two transition situations where non-uniqueness plays an important role. The first deals with the range -1 < m ≤ 0 in n = 1, which looks like supercritical but contains the non-uniqueness phenomenon for the Cauchy problem. The second is the study of logarithmic diffusion in the plane, i.e., the case m = 0 for n = 2 which has many appealing features for the analyst and the geometer.